Crystals

by A. E. H. Tutton

Fig. 70.—The Rhombohedron and its Axes.

But one alternate set of three faces of the hexagonal pyramid at one end, and the oppositely alternate set of three similar faces at the other end, will usually be found to be much less brilliant (indeed often quite dull) than the other alternate three, and very frequently also the amount of development is markedly different, both facts indicating that the terminal faces belong to two different but complementary rhombohedral forms, and that the system of symmetry is the trigonal and not hexagonal.

But there is much stronger evidence than this for trigonal symmetry. For the little faces marked s and x on Figs. 68 and 69 are characteristic of the trapezohedral class of the trigonal system, and it will be observed that on one crystal, Fig. 68, these faces occupy and modify a left-hand corner or solid angle on the crystal, while on the other crystal, Fig. 69, they occupy and replace a right-hand solid angle. Now, if a plate be cut out of the former crystal perpendicularly to the axis of the hexagonal prism, that is, to the optic axis of the trigonal uniaxial crystal, it will be found to rotate the plane of polarisation to the left, the direction in which the small faces are situated; while if a similar plate be cut out of the right-handed crystal shown in Fig. 69, that is, one which has the small faces on the right, it will be observed to rotate the plane of polarisation to the right.

As quartz possesses the symmetry of the trigonal system and is thus optically uniaxial, its optical properties are expressed, in common with those of all trigonal, tetragonal, and hexagonal crystals, by an ellipsoid of revolution, an ellipsoid the section of which perpendicular to the principal axis—that of revolution, the maximum or minimum diameter of the ellipsoid—is a circle. The optical properties are consequently the same in all directions round this axis, which has already been referred to by its common appellation of the “optic axis.”

The optic axis is identical in direction with the trigonal axis of symmetry in the case of quartz or other trigonal crystal, and in the cases of hexagonal and tetragonal crystals with the axes of hexagonal and tetragonal symmetry, these three axes of specific symmetry being the distinctive property of these three respective systems, which are thus known in common as optically “uniaxial.”

Consequently, no double refraction is suffered by a ray transmitted parallel to the optic axis, and the refractive index is equal in all directions perpendicular to the optic axis, that is, for all rays vibrating perpendicularly to the axis; hence the value of the refractive index obtained along any such direction is one extreme value for the whole crystal, and as already mentioned is distinguished by the letter ω. The refractive index along the direction of the axis itself is the other extreme value, and is labelled ε. It must be clearly appreciated, however, that it is not the direction of transmission but that of vibration perpendicular thereto, that is meant when it is said that, for instance, the direction of the axis corresponds to the index ε. That is to say, a ray the vibrations of which occur parallel to the optic axis of a uniaxial crystal is refracted to an amount which corresponds to the refractive index ε, while a ray the vibrations of which occur perpendicularly to the axis affords ω. The difference between ε and ω is the measure of the double refraction of the crystal.

In the case of quartz ε is the greater, being 1.5534 for sodium light, quartz being thus positive according to the convention already alluded to; while ω is the smaller, namely, 1.5443. In the case of the other widely distributed trigonally uniaxial mineral calcite, carbonate of lime CaCO₃, the opposite is the case, ω being the greater, having the value 1.6583 for sodium light, and ω the less, namely, 1.4864, calcite being thus a negatively uniaxial substance. The amount of the double refraction in the cases of the two minerals is very different, ε-ω for quartz being 0.0091, and ω-ε for calcite being as much as 0.1719. Calcite is indeed a mineral endowed with an especially large amount of double refraction, a property which renders it so eminently suitable for use in demonstrating the phenomenon, and for the construction of the Nicol polarising prism, in which one of the two mutually perpendicularly polarised rays, that which affords the index ω, is got rid of by total reflection at a balsam joint, a large rhomb of calcite being cut in half along a particular diagonal plane and the two halves cemented together again with Canada balsam; the other ray, which affords ε (but not at its minimum value), is transmitted as a beam of perfectly polarised light.

The result of this difference in the amount of the double refraction of the two minerals quartz and calcite is very interesting as regards their behaviour with polarised light. A thin plate of quartz, such as is often found in the slices of rock sections employed for microscopic investigation, of muscovite granite or quartz porphyry for instance, and which is usually about one-fiftieth of a millimetre in thickness, shows brilliant colours in a parallel beam of polarised light, the Nicol prisms of the polarising microscope being crossed for the production of the dark field before the introduction of the section-plate on the stage. This is only true, however, when the plate has not been cut perpendicularly to the axis, for such a thin plate thus cut does not perceptibly affect the dark field, there being no double refraction of rays transmitted along the axis, and the interference colours afforded by crystal plates in polarised light being due to the interference of the two rays produced by double refraction, one of which is retarded behind the other so as to be in a different phase of vibration. Also, the plate, even when cut obliquely, and best of all parallel, to the axis, has to be rotated in its own plane (perpendicular to the optical axis of the microscope), to the favourable position for the production of the most brilliant colour. This especially favourable position is halfway between (at 45° to) the positions at which darkness is afforded by the plate. For on rotating the plate between the crossed Nicols it becomes four times dark during a complete revolution, and at places exactly 90° apart, known as the “extinction positions,” whenever, in fact, that plane perpendicular to the plate which contains the optic axis is parallel to the plane of polarisation of either the polarising or analysing Nicol. At the intermediate 45° positions the maximum colour is produced.

The colour owes its origin, as already mentioned, to the interference of the two rays, corresponding to the two refractive indices, into which the light is divided on entering the crystal in any direction except along the axis. For one of the rays is retarded behind the other owing to the difference in velocity which is expressed reciprocally (inversely) by the refractive indices, and thus a difference of phase is produced between the two light-wave motions, with the inevitable result of interference when the vibrations have been reduced to the same plane by the analyser; light of one particular wave-length is then extinguished, and the plate therefore exhibits a tint in which the complementary colour to that extinguished predominates. The light which leaves the polarising Nicol is vibrating in one plane, but on reaching the crystal this is resolved into two rays vibrating at right angles to each other, and at 45° on each side of its previous direction of vibration, supposing the crystal to be arranged for the production of most brilliant colour. On reaching the analysing Nicol, the function of which is to bring the two vibrations again into the same plane, these two rays are each separately resolved back to the planes of vibration of the two Nicols, and that pair (one from each ray) vibrating parallel to the analysing Nicol are transmitted, while the other pair are extinguished. The two former rays thus surviving, one individual ray of the two having one refractive index and the other individual the other index, are thus in a position to interfere; for they are composed of vibrations in the same plane and of practically the same intensity, and differ only in phase. Extinction occurs when this amounts to half a wave-length, or an odd multiple of this, to which, however, requires to be added half a wave difference of phase which is introduced by the operation of the analyser. This explanation is a general one, applicable to thin plates of crystals belonging to all the six systems of symmetry other than the cubic. For plates of the latter, unless they are in an abnormal condition of strain, do not polarise.